If f(z)=u+iv is analytic and u-v=ex(cos y-sin y) find f(z) in terms of z.

If is analytic, then the Cauchy-Riemann conditions hold:



I'll denote these partial derivatives by .

You have



Adding and subtracting and , respectively, gives



Because is analytic, this system reduces to



Integrating with respect to gives




Differentiating with respect to gives



where is an arbitrary constant.

On the other hand, you have



(Note that and by the C-R conditions.) Integrating with respect to gives




Differentiating with respect to gives



It follows that





where the last equality is an invocation of Euler's formula.

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